All attracting local Morse data are isolated homotopic to each other, hence their local Morse homologies are isomorphic. The Morse cohomology of an attracting local Morse datum \(=(f,g,N,)\)  is isomorphic to the singular cohomology of \(N\) , i.e. \(HM^*()\cong H^*(N;R)\) , see for example [1]}, [2]}.
